1

An integrated lot-sizing model with supplier and carrier selection and

quantity discounts considering multiple products

S. J. Abrishamia, H. Vahdani

a,* and B. Rezaee

b

a. Department of Industrial Engineering, Faculty of Engineering, University of Bojnord, Bojnord, P. O. Box 94531-

55111, Iran

b. Industrial Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, P.O. Box

91775-1111, Iran

ABSTRACT

Because of growing competition in the global markets and the vital role of suppliers in business success,

the subject of supplier selection has attracted many researchers and practitioners during the recent years.

In addition to the supplier selection, the order quantity discount provided by the suppliers, is considered

through a new mixed-integer linear programming (MILP) model involving a manufacturer with multiple

products and multiple purchasing items over multiple periods. According to the proposed model, the

manufacturer purchases different amount of raw materials from selected suppliers in order to produce

different products. Customers’ demands are fulfilled by minimizing the total purchase, inventory,

production, and transportation costs over a multi-period planning horizon. Since the problem is NP-hard,

an efficient genetic algorithm (GA) is used to solve the large-scale real-world instances. The results are

compared with results from the exact approach wherever possible in order to investigate the efficiency of

the algorithm.

Keywords: Supplier selection, Inventory management, Quantity discounts, Carrier selection, Genetic

algorithm.

1. INTRODUCTION

Selecting the best supplier among a large number of suppliers sits on the top of virtually all

companies’ agendas. In practice, in the increasingly competitive market, indicators such as the

offered prices and quality significantly affect the selection of suppliers. The costs of raw

materials which were supplied and purchased, comprises more than 50% of all costs [1].

Indeed, there is a strong correlation between the supplier selection problem and other subjects

such as inventory management as well as transportation strategies. Regarding the inventory

management issues, the decision-making process becomes more complicated when suppliers

offer their discount conditions in order to obtain a larger market share which in turn, leads

customers to have the benefit of purchasing materials at reduced costs. The subject of quantity

discounts has attracted significant attention in the literature over several decades. In the field of

supplier selection and order allocation within the supply chain context, some researchers have

accounted for the benefits of quantity discounts in their presented models [2-9].

Paying special attention to the logistics and transportation costs by selecting the shipping system

properly, play a considerable role in the mentioned area of the study, although considering carrier

selection in the supplier selection literature is rare [10-12].

In this study we examine a supply chain seeking to select appropriate suppliers in order to reduce

the costs associated with inventory management, production management, and transportation, in

a dynamic planning environment. The multi-product MILP model which is developed in this

paper, considers the decision variables related to the selection of suppliers, the amount of raw

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material purchased from each supplier per period, carrier selection and inventory management

policy for each period. Accordingly, the presented model minimizes the total cost of supply chain

which includes raw materials, contracts, purchases from the supplier(s), production, storage and

carrier selection costs. To the best of our knowledge, considering all these factors simultaneously

has not received any attention in the literature. Specifically, we believe our main contribution is

to include the procurement of raw materials for present and upcoming production periods based

on the discount patterns and converting them to final products while dealing with a variety of raw

materials and products with different production plans, within an integrated supplier selection

model while the carier selection is taken into account, as well.

Since the modeling of the lot-sizing problems with above-mentioned assumptions belongs to the

family of NP-hard problems [13, 14], an efficient GA is provided. In order to validate the

proposed algorithm, it is implemented and analyzed on two data sets incorporating 20 samples,

one consisting of small-scale sample problems with obtainable optimal solutions, and the other

large-scale ones which their optimal solutions can not be reached through the exact method. A

direct comparison is made based on the value of objective function and elapsed runtime.

The remainder of this paper is organized as follows. Section 2 presents a brief review of the

existing relevant quantitative approaches related to the supplier selection and order allocation

problems along with the solution methods. Section 3 determines the mathematical model and

examines its validation on the small-scale examples based on several scenarios. In section 4, the

proposed GA is presented and applied on twenty small/large-scale samples and the results are

compared with the exact approach. Finally, conclusions and suggestions for the future research

are presented in section 5.

2. LITERATURE REVIEW

The supply of inexpensive materials from high-quality sources has great impact on developing

successful business strategies. For extensive literature reviews on “supplier selection and

decision making”, we can refer to De Boer et al. [15], Ho et al. [16] and Chai et al. [17]

consisting of different decision-making techniques such as Multi-Criteria Decision Making

(MCDM), Mathematical Programming (MP), and Artificial Intelligence(AI) along with

integrated approaches (AHP, ANP, DEA etc.).

Ghodsypour and O’Brien [18] proposed a mixed integer non-linear programming (MINLP)

model with multiple sources which consider the net inventory holding price as well as the

transportation and ordering costs. Basnet and Leung [19] proposed a supplier selection and an

inventory lot-sizing model including multiple products in a multi-period horizon to minimize the

purchasing, ordering and inventory holding costs. Liao and Rittscher [10] investigated a

nonlinear multi-objective model that includes minimizing the cost of logistics and transportation,

defective items and delivery time objectives along with the selected carrier. Rezaei and Davoodi

[13] presented two multi-objective mixed integer non-linear programming (MOMINLP) models

for multi-period lot-sizing supplier selection problems consisting of multiple products with

shortage and non-shortage assumptions. Their objectives were to optimize the cost (purchasing,

ordering, holding and transportation), quality, and service level. In another study, they proposed a

MINLP model with three objectives of total profit, inconsistency and deficiency [20]. They used

NSGA-II to solve their model. Mendoza and Ventura [21] proposed two mixed integer nonlinear

programming models to select the best set of suppliers and to determine the proper allocation of

order quantities. Their aim was to minimize the annual ordering, inventory holding, and

purchasing costs under suppliers’ capacity and quality constraints. Senyigit [22] presented a

MILP lot-sizing supplier selection model. His model incorporated a multi-product, multi-echelon

defective supply chain network seeking to minimize the total relevant costs. Aliabadi et al. [23]

provided an integrated non-linear binary model with multiple items for the supplier selection

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problem. In their study, inventory costs for both suppliers and buyers, production costs for

suppliers and transportation costs are considered. Ventura et al. [24] provided a multi-period

inventory lot-sizing model for a single product in a serial supply chain. The objective of their

MINLP model was to minimize the total variable costs including purchasing, production,

inventory, and transportation costs. They also considered the all-unit quantity discounts approach

for the actual transportation costs.

A stochastic lot-sizing problem with multiple suppliers along with quantity discounts was

presented by Kang and Lee [25]. Their multi-objective programming model consisted of cost

minimization and service level maximization functions. They used an efficient heuristic dynamic

programming model to solve the MILP model. A MINLP model was presented for the order

allocation by Meena and Sarmah [8] which considered quantity discounts for potential suppliers

solved by a GA approach. In their research, the suppliers are exposed to the risk of failure due to

man-made or natural disruptions. Gorji et al. [26] presented a two-level supply chain with one

retailer and several suppliers involving multiple periods to determine both supplier selection and

inventory allocation policies. Total profit was the objective function of their MINLP model

which was solved by GA. Lee et al. [14] provided a MIP model for the lot-sizing supplier

selection problem encompassing multiple periods along with all-unit and incremental discounts.

They sought to minimize the costs including ordering, holding, purchasing, and transportation via

an efficient GA. Choudhary and Shankar [11] represented a single-objective MIP model that

considered supplier selection along with carrier selection. They allowed shortage on a single

product and used the exact method to solve the model. In their latest research, Choudhary and

Shankar [12] presented a goal programming model for inventory lot-sizing with the supplier

selection and carrier selection decision process. Their single-product multi-objective integer

linear programming (MOILP) approach incorporated the all-unit quantity discounts which were

offered by potential suppliers. Cardenas-Barron et al. [27] presented a new reduce and optimize

approach to solve the MILP for the multi-product multi-period inventory lot sizing with supplier

selection problem solved by a heuristic algorithm. Zaheri et al. [28] proposed two models to

formulate a Supplier selection in a single-buyer, multi-supplier two-echelon supply chain

network under all-unit quantity discount policy that both buyer and vendor play the leadership

role in a bi-level programming model structure. Ghaniabadi and Mazinani [29] presented the

single product dynamic lot sizing problem with supplier selection and backlogging in the

presence of incremental and all-units quantity discounts. Alfaresa and Turnadi [30] provided a

MIP model for a realistic multi-item lot-sizing problem with multiple suppliers, multiple time

periods, quantity discounts, and backordering of shortages. In order to tackle larger problem

sizes, two heuristic solution methods are proposed including a Silver-Meal heuristic and a

problem-specific GA. Suriyan and Rungreunganun [31] present mathematical developments

minimizing the total costs of inventory management MILP and supplier selection for inventory

lot-sizing problem with quantity discounts with multi-products through multi-periods with the

consideration of supplier lead-times and under the conditions of purchasing budgets.

Kanchanaruangrong and Woarawichai [32] proposed a multi-product and multi-period inventory

lot-sizing problem with supplier selection under vehicle capacity and both all-units and

incremental quantity discounts. The buyer’s objective of their MILP model was to minimize

costs including purchase cost of the products, ordering cost for the suppliers, transportation cost

for the suppliers and holding cost of the remaining inventory in each period. Zaheri et. Al [33]

aimed to formulate a supply chain with a buyer and multiple suppliers using bi-level

programming approach with a hierarchical structure and PSO and Differential Evolution (DE)

algorithms were developed to solve their nonlinear bi-level programming problem (NBLP).

Table 1 illustrates a comparison between the present research and other existing studies in this

area.

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(Table 1 should be inserted here)

In the supplier selection domain, there are a few studies that have simultaneously considered

quantity discounts while shipping through the selected carriers. Most of these studies focus on

single product mode and do not account for the production constraints. As a result, raw materials

and time constraints, which are the most important components in a production process, have not

been examined in the literature. The purpose of this study is to support decision makers by

providing a MILP model looking to establish a trade-off among inventory holding, production,

transportation and procurement process costs, within an all-unit quantity discounts environment.

3. PROBLEM DEFINITION AND MODEL DEVELOPMENT

With the expansion of the competitive atmosphere, it is remarkable that in the real world each

manufacturer can satisfy their demands through different suppliers providing different quantity

discounts and then carry them using various carriers. According to this, we consider a supply

chain containing several suppliers which supply different raw materials and a manufacturer that

produces different types of products to fulfill the customers’ demands, so that raw materials are

transported to manufacturer by different transportation carriers. The total cost is the sum of the

purchasing, ordering, production, inventory holding and transportation costs. The solution to the

problem determines which raw materials should be ordered, in what quantities, from which

supplier(s), by which transportation carrier(s), and during which time periods, so that the

customers’ demand is satisfied whereas the product shortage is not allowed. There is also the

possibility for the manufacturer to take advantage of discounts to purchase raw materials at large

volumes and store them to fulfill the customers’demands in the future. Due to the production

costs and time constraints in each period, the manufacturer should adopt an efficient policy on

purchase quantities in order to reach a cost balance. In the meantime, the costs of shipment

between the manufacturer and customers are undertaken by customers; thus, it is not considered

in the model. Fig.1 shows the assumed supply chain network.

(Figure 1 should be inserted here)

Other assumptions considered in this study are as follows:

The demand quantities for multiple products are known but inconstant over the multiple

planning periods.

The price of unit raw material received from suppliers depends on the order quantity in

each period. With this in mind all-unit quantity discount is applied.

Inventory holding costs are known for raw material and product and vary in each period.

The planning horizon is finite and known. The length of each period is fixed as well.

Transportation costs depend on the selected carrier(s), which in turn depend on the

selected supplier and period.

The supply capacity of suppliers is finite and different in each period.

The storage capacity of raw materials and products is finite and different in each period.

Each carrier has a distinct capacity which is limited.

The number of available carriers is finite and different in each period.

The notations of the model are presented as follows:

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Set indices:

i the index of the raw materials purchased from the suppliers (i = 1, 2, …, I)

j the index of the suppliers (j = 1, 2, …, J)

k the index of the all-unit price break levels (k = 1, 2, …, K)

t the index of the discrete planning periods (t = 1, 2, …, T)

l the index of the transportation carriers (l = 1, 2, …, L)

p the index of the products (p = 1, 2, …, P)

Parameters:

Dpt the total demand of the p-th product in period t

Ojt the cost of placing order to supplier j in period t

PRpt the production cost per unit of product p in period t

CPt the storage capacity of the products of the manufacturer in period t

tp the time needed to produce one unit of product p

Tt total time available for production in period t

αip the units of raw material i required to produce one unit of product p

CRt the storage capacity of the raw materials of the manufacturer in period t

CSijt the supply capacity of raw material i by supplier j in period t

Wi the cubic volume of raw material i

UVl the maximum volume capacity of carrier l

UNVlt the maximum available number of carrier l in period t

Bijkt the quantity at which all-unit price break k occurs at supplier j in period t for raw material

i

Pijkt the cost of procuring one unit of raw material i from supplier j at price break level k in

period t

hit the inventory holding cost per unit raw material i during period t

Hpt the inventory holding cost per unit product p during period t

CVjlt the transportation cost from supplier j by carrier l in period t

M a sufficiently large number.

Decision variables:

Xijktl the quantity of raw material i purchased from supplier j at price break level k in period t

transported by carrier l

Ypt the quantity of product type p manufactured in period t

NVjlt the quantity of carrier l used in period t for supplier j

zjt a binary variable: 1if supplier j is chosen in period t, 0 otherwise

vjlt a binary variable: 1 if carrier l is used in period t for supplier j, 0 otherwise

Iit the inventory level of raw material i at the end of period t

I'pt the inventory level of product p at the end of period t

bijkt a binary variable: 1 if the lot-size of raw material i purchased from supplier j placed in

price break level k in period t, 0 otherwise.

3.1. Problem formulation

According to the above parameters, decision variables, and model assumptions, the mathematical

model can be formulated as follows:

𝑀𝑖𝑛 𝑍 = 𝑍1 + 𝑍2 + 𝑍3 + 𝑍4 + 𝑍5 (1)

𝑍1 = ∑ ∑ ∑ ∑ ∑ Xijktlltkji Pijkt (1a)

𝑍2 = ∑ ∑ zjt Ojt𝑗𝑡 (1b)

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𝑍3 = ∑ ∑ 𝑌pt𝑃𝑅pt𝑝𝑡 (1c)

𝑍4 = ∑ ∑ Iitit hit + ∑ ∑ I'ptpt 𝐻pt (1d)

𝑍5 = ∑ ∑ ∑ CVjltljt NVjlt (1e)

Subject to:

𝐼′pt = 𝐼′

p(t-1) + 𝑌pt − 𝐷pt ∀ 𝑝 , 𝑡 > 1 (2)

∑ 𝐼′pt ≤ 𝑝 𝐶𝑃t ∀ 𝑡 (3)

∑ 𝑌pt 𝑡p𝑝 ≤ 𝑇t ∀ 𝑡 (4)

𝐼it = 𝐼i(t-1) + ∑ ∑ ∑ 𝑋ijktl𝑙 − ∑ 𝑌pt𝑝𝑘𝑗 𝛼it ∀ 𝑖 , 𝑡 > 1 (5)

∑ 𝐼it𝑖 ≤ 𝐶Rt ∀ 𝑡 (6)

∑ ∑ Xijktllk ≤ CSijt Zjt ∀ 𝑖 , j , t (7)

∑ 𝑣jlt𝑙 = zjt ∀ j , t (8)

∑ ∑ 𝑋ijktl𝑘𝑖 ≤ 𝑀 𝑣jlt ∀ i , j , k , t , l (9)

∑ ∑ 𝑋ijktl 𝑊i𝑖𝑘 ≤ 𝑁𝑉jlt 𝑈𝑉l + 𝑀 (1 − 𝑣jlt) ∀ 𝑗 , l , t (10)

∑ 𝑁𝑉jlt𝑗 ≤ 𝑈𝑁𝑉lt ∀ l , t (11)

Bij(k-1)t bijkt ≤ ∑ Xijktll ≤ Bijkt bijkt ∀ i , j , t , k >1 (12)

∑ 𝑏ijkt𝑘 = 𝑧jt ∀ i , j , t (13)

Iit . I'pt ≥ 0 ∀ i , t , p (14)

𝑋ijktl , 𝑌pt , 𝑁𝑉jlt ≥0 𝑎𝑛𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 ∀ i , j , k , t , l , p (15)

𝑧jt , 𝑣jlt ,bijkt 𝜖 {0 , 1} ∀ 𝑖 , j , k , l , t (16)

The objective function (1) is to minimize the total cost over the planning horizon. It consists of

five terms: (1a) the costs of the purchase, (1b) the ordering costs, (1c) the production costs, (1d)

the inventory holding costs for the remaining inventory at the end of each period, which includes

raw materials and products holding costs respectively, and (1e) the transportation costs.

Constraint (2) represents the inventory balance of products in different periods. Constraint (3)

enforces the products’ storage capacity per period. Constraint (4) controls the capability of

production in each period by time limit production.Constraint (5) computes the amount of raw

materials that remain at the end of each period. Constraint (6) determines the raw materials’

storage capacity per period. Constraint (7) ensures that raw materials will be provided through

suppliers’ contracts with the manufacturer. The purchase quantity must not exceed the maximum

supplier capacity. Constraint (8) indicates that if the contract is performed by a supplier, a

transportation system must also be considered. Constraint (9) assures that if the carrier is not

assigned to a supplier, the supplier will not be able to ship the raw materials. Constraint (10)

determines the number of carriers used for shipping with respect to its volume. Constraint (11)

guarantees that the total number of carriers deployed for transportation will not surpass the

maximum number of carriers. Constraints (12-13) are related to the all-unit quantity discount

plan. According to these constraints, each lot is purchased from suppliers at a specified price

break in the discount interval. Only one price break level which is applied to the lot-size in each

period and supplier is considered. Finally, Constraints (14-16) identify the type of different

variables.

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3.2. Numerical example

To explain the performance of the proposed mathematical model, a numerical example is

provided in which the manufacturer wants to purchase three types of raw materials to produce

two types of products over a planning horizon of five periods through three potential suppliers.

Customers’ demands are 20 and 30 for the product number one and two in each period,

respectively. The coefficient of raw materials usage for production, the volume of raw materials

as well as the capacity of suppliers- which have been considered to be constant through different

periods- are given in Table 2. The ordering costs are assumed to be constant in each period,

which are 120, 100 and 110 in a monetary unit, respectively. Inventory holding costs for each

raw material unit are 2, 3, and 2 in a monetary unit, respectively. Inventory holding costs of

products, the time needed to produce one unit of each product and production costs are shown in

Table 3. The total production time available in each period is 500 time units. The Storage

capacities for the raw materials and products in manufacturer area are 1000 and 100 per period,

respectively.

The suppliers’ proposed prices for the raw materials considering the price break levels are

presented in Table 4. In this example, the discount levels for different periods and raw materials

are assumed to be identical.

(Table 2, 3 and 4 should be inserted around here)

Table 5 shows the transportation costs proposed by the suppliers considering the carriers and

their maximum capacity volume. Moreover, it displays the maximum number of available

carriers, which varies by the period.

(Table 5 should be inserted here)

Table 6 shows the solution obtained from solving the model. As the results show, the optimal

solution is to purchase the first raw material (400 units), the second raw material (450 units), and a

fraction of the third one (200 units) from the first supplier. The second supplier is selected to

supply 300 units of the third raw material, because of the better discount and closer distance of

supplier 1, which is obvious in table 6.

(Table 6 should be inserted here)

3.3. Scenarios

For a more comprehensive assessment of the problem parameters, different scenarios have been

examined. In this section, we assess two important scenarios to delve into the impact of different

quantity discounts and inventory costs on the objective function, while keeping the rest of the

parameters at their base value. Three additional scenarios assess the effects of quantity discounts

and five further scenarios investigate the inventory costs for the raw materials and products.

3.3.1. The analysis of quantity discounts

As mentioned before, to analyze the effects of quantity discounts, three sub-scenarios are defined

and provided in Table 7. In sub-scenario 1, the first supplier does not offer any discounts,

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whereas suppliers 2 and 3 offer their previous discounts. As expected, we observe that most of

the materials (over 82% of all purchases) are now sourced from suppliers 2 and 3. Thus,

manufacturer’s purchase from supplier 1 is decreased by about 60%.

In the second sub-scenario, only supplier 3 offers quantity discounts and as a result, a significant

percent of the purchases (59%) is made from supplier 3. In addition, because of the proximity of

supplier 1, the remaining purchases are sourced from supplier 1.

In sub-scenario 3, suppliers do not offer any discounts, and the selection is made based on other

problem criteria. Due to more appropriate conditions offered by the first supplier, all purchases

are allocated to it.

(Table 7 should be inserted here)

Fig. 2 illustrates the effects of these variations on the costs. As expected, by eliminating the

quantity discount advantage, the total costs as well as the purchasing costs increase, but the costs

of ordering and transportation do not show any significant difference. The model strives to find

the best trade-off among these cost categories to compensate the increase in the purchasing costs.

(Figure 2 should be inserted here)

3.3.2. The analysis of inventory holding costs

Table 8 illustrates how the amount of purchases is affected by the inventory holding costs. There

are five scenarios in which the effects of changes on inventory holding cost for the products and

the raw materials are examined. In scenarios 1 and 2 the inventory holding cost for the products

is decreased and increased, respectively.

In scenario 1, the reduction in the holding costs of products drives the manufacturer to supply the

raw materials and produce the final products less frequently and in larger sizes in order to take

advantage of the offered discounts. From Table 8, this practice reduces the total ordering cost,

whereas the total inventory of the products at the end of each period will significantly increase.

In scenario 2, as expected, the significant increase in the products’ inventory holding costs leads

to zero inventory level for the products at the end of each period and an increase in the number of

orders.

In scenario 3, as the inventory holding costs of the raw materials decreases, the manufacturer

supplies the raw materials in larger quantities, which in turn decreases the inventory level of the

final products at the end of each time period. This policy will also provide the manufacturer with

more opportunities to take advantage of the offered discount privileges. In Scenario 4, as the

inventory holding costs of raw materials increases, the objective function increases, whereas the

decision variables remain unchanged compared to that of the basic example. The latter is a direct

effect of the existing trade-off between the potential discount benefits of ordering in larger

quantities versus the rise in the holding costs, which prompts the manufacturer to follow its

existing course of ordering and production policies. In Scenario 5, the raw materials undergo a

substantial increase in the holding costs to the extent which cannot be offset by the suggested

discount benefits. Table 8 shows the change in the order and production policies, so that the

inventory only comprises the final products, i.e., no raw material inventory is held at the end of

any time period.

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In a nutshell, our analyses show that the costs related to raw materials and the final products have

consequential influences on supply chain policies, performance, and costs. Therefore, we

recommend that supply chain players should acquire more accurate business cost estimations.

(Table 8 should be inserted here)

4. SOLUTION METHOD

In the literature, because of the NP-hardness of lot-sizing in this area, different methods have

been used to solve the supplier selection problem. The results of applying GA to a wide variety

of supplier selection problems have been very promising among other evolutionary algorithms in

both single and multi-objective problems (Liao and Rittscher [10]; Rezaei and Davoodi [13, 20];

Lee et al. [14]; Aliabadi et al. [23]; Gorji et al. [26]). Therefore, encouraged by its noticeable

positive features, a GA is proposed in this study to handle large scale problem instances.

4.1. Chromosome representation

In order to solve the problem, a chromosome is designed with four parts. The first part of the

chromosome is a P×(T-1) matrix which is used to calculate the quantity of the leftover products

after consumption in each period. The elements of this matrix belong to the standard normal

distribution function. Since the premise of the problem is to fulfil customer demand in each

period, the available inventory level of the products at every period is determined in this part of

the chromosome. The second part of the chromosome is an I×(T-1) matrix representing the

amount of the raw materials left after consumption in each period. The elements of this matrix

also have a standard normal distribution. The amount of the raw materials needed in each period

is calculated by the following equation:

𝑇ℎ𝑒 𝐷𝑒𝑚𝑎𝑛𝑑 𝑜𝑓 𝑟𝑎𝑤 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑖 𝑛𝑒𝑒𝑑𝑒𝑑 𝑖𝑛 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡 = ∑ 𝑌pt𝑝 𝛼it ∀ 𝑝 , 𝑡 > 1 (17)

With regard to the values of the second part of the chromosome, the purchasing amount of the

raw materials can be calculated by the following equations in each period:

𝑇ℎ𝑒 𝑃𝑢𝑟𝑐ℎ𝑎𝑠𝑖𝑛𝑔 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓𝑟𝑎𝑤 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑖 𝑖𝑛 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡= 𝑇ℎ𝑒 𝑑𝑒𝑚𝑎𝑛𝑑 𝑜𝑓 𝑟𝑎𝑤 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑖 𝑖𝑛 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡× [1 + 𝑇ℎ𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 𝑚𝑎𝑡𝑟𝑖𝑥 (𝑖,𝑡)]− 𝑇ℎ𝑒 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑝𝑒𝑟𝑖𝑜𝑑 (𝑓𝑜𝑟 𝑡 < 𝑇)

(18)

and,

𝑇ℎ𝑒 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑖𝑛𝑔 𝑎𝑚𝑜𝑢𝑛𝑡𝑠 𝑜𝑓 𝑟𝑎𝑤 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑖 𝑖𝑛 𝑝𝑒𝑟𝑖𝑜𝑑 𝑇= 𝑇ℎ𝑒 𝑑𝑒𝑚𝑎𝑛𝑑 𝑜𝑓 𝑟𝑎𝑤 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑖 𝑖𝑛 𝑝𝑒𝑟𝑖𝑜𝑑 𝑇− 𝑇ℎ𝑒 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑝𝑒𝑟𝑖𝑜𝑑

(19)

After determining the required purchasing amount of raw materials by the first two parts of the

chromosome, the third part determines how to provide the purchase quantities belonging to each

supplier by an i×j×t matrix with elements taking values from [0,1] and each corresponding to a

supplier and a raw material in a period. The elements of this matrix show the supply priority of

the specific raw material by the specific supplier in each period. The chromosome numbers are

used in compliance with the supplier capacity constraints in a way that the supplier with the

highest priority is selected to provide the maximum amount of raw materials. The other suppliers

10

will be selected in the order of precedence to meet the total demand. In the fourth part of the

chromosome, carriers are allocated to each selected supplier. This part is a j×l×t matrix with

[0,1] element values and is used to cover the carriers’ capacity constraints.

4.2. Initialization of the population

There are different approaches to produce the initial population which is the starting point of the

algorithm. In this study, we have adopted the random generation of solutions approach to create

the initial population. In this way, for each population, a chromosome with the specifications

described in the previous subsection is randomly generated. The feasibility conditions that must

be assessed after generating each chromosome are:

Limitation of time availability for production

Product storage capacity constraints

Storage capacity constraints for the raw materials.

4.3. Fitness function

Each chromosome is a plausible solution of the problem which is used to calculate the fitness

value of the objective function. Since the objective function is based on costs, smaller objective

function values generate higher fitness values.

4.4. Selection strategy

The solutions of GA evolve from each generation to the next. Evolution of the population is

realized through GA operators such as crossover, mutation and elitism. The hatching process in

which we need to select the solutions (chromosomes) from the population, is accomplished

through a crossover operation. Different methods have been proposed to select the parents

(boltzmann selection, tournament, rank, steady state etc.). The selection method by the Roulette

wheel which has been used in this study is among the most popular ones. Moreover, the elitism

idea has been implemented on the GA and a percentage of the best chromosomes in the current

population will be kept for the next generation.

4.5. Crossover operation

The crossover operator is implemented based on a guide matrix with binary elements. For each

chromosome, there is a guide matrix with equal dimensions. To generate new offsprings, the

crossover operator is applied to the parents selected by the Roulette wheel approach. If the value

of the corresponding element in the guide matrix equals 1, the corresponding values of the parent

element would be replaced, otherwise the element is left unchanged. Fig. 3 represents an example

of a crossover operation.

(Figure 3 should be inserted here)

4.6. Elitism operation

According to this operator, the chromosomes having the best fitness value in the active

generation are automatically carried over to the next. This operator causes further improvement

in the efficiency of GA by preventing the loss of good solutions. In other words, the solution

quality obtained by the GA will not decrease from one generation to the next.

11

4.7. Local search

In each iteration of the provided GA, we search to find better neighbourhoods around better

solutions in the population. A certain percentage of the best solutions for each generation of the

algorithm will be examined in order to find better neighbourhoods. To create a neighbourhood

for the algorithm in every part of the chromosome, new elements are calculated through the

following equation:

New element = (1 + R) * Old element

In this equation, “R” is a random number with the standard normal distribution. If the new

element takes a value more than 1, it will set to 1 and if it takes a value less than zero, it will be

considered equal to zero. Searching to find a better neighbourhood stops after finding a better

solution. Otherwise, after certain steps and finding no better solution, it stops.

4.8. Mutation operation

In the mutation operation, we randomly switch rows or columns of the chromosome. For the

parts of the chromosome that has more than one dimension, this action is applied to each and

every dimension. Fig. 4 shows an example of a mutation operation.

(Figure 4 should be inserted here)

4.9. Termination criterion

The most common termination criteria in GA include the fixed number of iterations (or a fixed

length of computational time), an arbitrary number of iterations where no further improvement is

realized in the objective function, and when the predefined threshold in the objective function

value is reached. As for the proposed design, we let the iterations proceed to a specified number;

however, the algorithm terminates if no improvement is achieved during 50 consecutive

iterations.

4.10. Parameter setting of the proposed GA

We have borrowed the same parameters setting method form Gorji et al [26] and Dai and Zheng

[34], based on performing considerable amount of experimental analyses. To find the optimal

solution efficiently, some parameters of the GA require to be set appropriately, including the

population size, maximum generation number, crossover rate, mutation rate, elitism rate as well

as the local search rate. According to this statement, these parameters will be set to achieve the

best optimal minimum cost possible in a short time.

Population size. It is noteworthy that if the population size is huge, it would probably lead to

very slow rate of convergence and elude sub-optimal solutions. On the other hand, a small

population probably results in a premature convergence and quick rate of convergence. Thus,

in the presented article, two choices are available for the population size: 250 and 350 which

have moderately resulted in better solutions through extensive experiments having been done

on different values for this parameter.

The maximum generation number. Two options have been applied here: 150 and 200.

Crossover and mutation rate. Crossover and mutation are the ways to generate new

individuals which are mostly used to provide exploitation and exploration, respectively. In

this paper, each of them has two choices: 0.85 and 0.93 for crossover, 0.1 and 0.18 for

mutation.

12

The elitism rate. Elitism guarantees that the solution quality obtained from the GA will not

decrease from one generation to the next. In this paper, two choices have been considered

which are 0.1 and 0.25.

The local search rate. Two options have been applied here: 0.1 and 0.2.

As can be seen in Table 9, based on our computational experiments the following combination

obtains the best value for the objective function with the shortest time: the population size,

maximum generation, crossover rate, mutation rate, elitism rate and the rate of the local search

are set at “350”, “150”, “0.93”, “0.1”, “0.25”, and “0.2”, respectively. Of course, these

parameters can not guarantee a global optimal solution.

(Table 9 should be inserted here)

5. THE RESULTS OF THE NUMERICAL EXAMPLES

In this section, 10 small and 10 large sample problems are presented to examine the problem in

different scales. The experiments were performed on a PC with an Intel dual core i5 CPU at 2.5

GHz and 4 GB of RAM. The results obtained from the GA (coded in MATLAB software) are

compared with those of the exact method using LINGO 12. Twenty samples with different

dimensions illustrated in Table 10 are considered and the values of the common parameters

which are used in these samples are shown in Table 11.

(Table 10 and 11 should be inserted here)

The results which are obtained on small-scale and large-scale problems are presented in Table 12

and 13, respectively. The average run-time is derived from the average of five-time

implementations of the algorithm. The gap is obtained through comparing the averages of the

best solution values among those five-time implementations with that of the exact method. Fig. 5.

demonstrates the values of the objective function over small-scale samples.

(Table 12 and 13 should be inserted here)

As the dimension of the problem increases (samples 11 to 20), the exact method fails to reach the

optimal solution in a reasonable time while the proposed GA achieves near-optimal solutions at a

fairly short run and with a little variation. It is noteworthy that for samples 19 and 20, the exact

method fails to find a feasible solution even in 2,000 seconds and hence, we have terminated it in

4,000 seconds of run time.

(Figure 5 should be inserted here)

Considering Fig. 6, it can be observed that the run times for the GA are more than the exact

method for samples 1 to 8, while in the rest of the samples, this trend has changed showing the

effectiveness of the presented GA. Fig. 7 shows the rate of change in the objective function value

in samples 11 to 20, so that the gap of the solutions is very close which is a persuasive reason to

provide proof of the effectiveness of the presented GA.

13

(Figure 6 and 7 should be inserted here)

6. CONCLUSION

Today, due to the increasing demands and intense competition among product manufacturers, the

need for multi-product productions along with an efficient inventory management is felt more

than ever. This paper presents a multi-product MILP model for the supplier selection over a

multi-period planning horizon. The objectives include the purchase cost of the raw materials

from the suppliers, ordering costs, production costs, inventory holding costs of the raw materials

and products, as well as the transportation costs. In the problem setting, the manufacturer is

allowed to purchase the required raw material through the all-unit quantity discount plan offered

by the suppliers which are required to completely satisfy customers’ demands. Different raw

materials could be shipped by different carriers. The Proposed model seeks to determine which

raw materials should be ordered, in what quantities, from which suppliers, by which

transportation carriers, and in which periods. To solve the proposed model, an efficient GA has

been proposed and twenty small-scale and large-scale samples are presented to show the

applicability of the approach. The results are compared with those of the exact method to show

the performance of the proposed algorithm.

For the future research, the present model can be further developed by taking into account other

assumptions such as the backlog or loss in demand fulfilment, introducing probabilistic and fuzzy

parameters. In addition, other meta-heuristics can be employed to improve the quality of

solutions and reduce the run time needed to solve the problem. Introducing multiple competing

objectives to the problem setting can also serve as an interesting direction for the future research.

There are some other related areas that can be inspiring to develop this work such as: sustainable

supplier selection, green supplier selection and implementing the MADM approach to cope with

new problems regarding various optimization criteria [35-36].

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supplier selection under quantity discount policy”, Scientia Iranica E, 24(4),

pp. 2095-2104 (2017).

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backlogging and quantity discounts”, Computers and Industrial Engineering,

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32. Kanchanaruangrong, N. and Woarawichai, C. “Modeling and solving of multi-

product inventory lot-sizing with supplier selection under quantity discounts”,

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(2018).

33. Zaheri, F., Zandieh, M., Taghavifard, M. T. et al., E. “A bi-level programming

model to solve supplier selection and lot-sizing problem addressing quantity

discounts and transportation cost”, Industrial Engineering and Management

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Systems, 17(2), pp. 267-280 (2018).

34. Dail, Z. and Zheng, X. “Design of close-loop supply chain network under

uncertainty using hybrid genetic algorithm: a fuzzy and chance-constrained

programming model”, Computers and Industrial Engineering, 88, pp. 444-457

(2015).

35. Hajikhani, A., Khalilzadeh, M. and Sadjadi, .S. J. “A fuzzy multi-objective

multi-product supplier selection and order-allocation problem in supply chain

under coverage and price considerations: An urban agricultural case study”,

Scientia Iranica E, 25(1), pp. 431-439 (2018).

36. Rabieh, M., Rafsanjani, A. F., Babaei, L. et al.. M. “Sustainable Supplier

Selection and Order Allocation: An Integrated Delphi Method, Fuzzy TOPSIS

and Multi-Objective Programming Model”, Scientia Iranica E, Doi:

10.24200/SCI.2018.5254.1176 (Article in Press).

Salman Jameh Abrishami (S. J. Abrishami):

E-mail: salman_abrishami@yahoo.com, Tel.: +98 915 244 8682

Department of Industrial Engineering, Faculty of Engineering, University of Bojnord, Bojnord,

P. O. Box 94531-55111, Iran

Biography:

Salman Jameh Abrishami received his B.S. and M.S. degrees in Industrial Engineering from

“Sadjad University of technology” and “University of Bojnord”, Iran, 2012 and 2015,

respectively. Currently, he is a Ph.D. candidate of industrial engineering at Imam Hossein

Comprehensive University, Tehran, Iran in the direction of Mega Systems Analysis.

Dr. Hashem Vahdani (H. Vahdani): Corresponding author

.

E-mail: hvahdani@ub.ac.ir, Tel.: +98 912 276 8426; Fax:+98 58 324 10700;

Department of Industrial Engineering, Faculty of Engineering, University of Bojnord, Bojnord, P. O. Box

94531-55111, Iran

Biography:

17

Hashem Vahdani is an Assistant Professor at University of Bojnord, Bojnord, Iran. He holds

B.S., M.S., and Ph.D. in Industrial Engineering from Sharif University of Technology, University

of Tehran, and Sharif University of Technology in Iran, respectively. His research interests cover

applied operations research, production management, supply chain management with special

focus on sustainable development and green chains and warranty analysis.

Dr. Babak Rezaee (B. Rezaee):

brezaee@um.ac.ir, Tel.: +98 912 643 8902

Industrial Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad,

P.O. Box 91775-1111, Mashhad, Iran

Biography:

Babak Rezaee is an Assistant Professor in the Industrial Engineering department at Ferdowsi

University of Mashhad (FUM). His research interests lie in utilizing Artificial Intelligence and

Operations Research to tackle decision-making problems area. Some of the applications he has

worked on are Supply Chain Risk Management, Energy Policy and Modeling, Modeling

Uncertainty in Operations and Services and Designing Fuzzy Rule-Based Systems.

18

List of Captions:

a- Figure captions:

- Fig. 1. Supply chain network.

- Fig. 2. Effects of quantity discounts on the costs

- Fig. 3. An example of crossover operation

- Fig. 4. An example of mutation operation

- Fig. 5. The objective function values for small-scale samples

- Fig. 6. Run time comparison between GA and the exact method

- Fig. 7. The objective function values for large-scale samples

b- Table Captions:

- Table 1. Literature review summary on inventory lot-sizing with supplier selection

- Table 2. Data of the raw materials and suppliers’ capacities

- Table 3. Manufacturer’s data

- Table 4. Suppliers’ dataset for the numerical example

- Table 5. Transportation data

- Table 6. Results for base example

- Table 7. The impact of quantity discount

- Table 8. The impact of inventory holding costs

- Table 9. Best value and time for different parameters

- Table 10. Samples’ dimensions

- Table 11. The values of the parameters used in the samples

- Table 12. The computational results of small-scale samples

- Table 13. The computational results of the large-scale samples

19

Figures:

Fig.1. Supply chain network

Fig. 2. Effects of quantity discounts on the costs

25,055 26,575 27,353 27,465

17,050 17,735 18,755 20,750

460 650 470 600

3,825 4,550 4,630 3,450

0

5,000

10,000

15,000

20,000

25,000

30,000

B A S E E X A M P L E S U B - S C E N A R I O 1 S U B - S C E N A R I O 2 S U B - S C E N A R I O 3

CO

STS

Objective value

Purchasing cost

Ordering cost

Transportationcost

20

Fig. 3. An example of crossover operation

Fig. 4. An example of mutation operation

Fig. 5. The objective function values for small-scale samples

0

5000

10000

15000

20000

25000

30000

1 2 3 4 5 6 7 8 9 10

CO

ST

SAMPLE

LINGO GA

21

Fig. 6. Run time comparison between GA and the exact method

Fig. 7. The objective function values for large-scale samples

1 1 2 3 5 5 7

20

134

123

11 13

34 34

27

47 53

62

87

119

1 2 3 4 5 6 7 8 9 10 Sample

Time (Lingo) Time (GA)

75000

125000

175000

225000

11 12 13 14 15 16 17 18 19 20

CO

ST

SAMPLE

LINGO GA

22

Tables:

Table 1. Literature review summary on inventory lot-sizing with supplier selection

Study Formulation Objective Constraints Quantity

discounts

Transportation

cost

Carrier

selection

Multi-

product Remarks

Mendoza

& Ventura

[21]

MINLP

Two model for minimizing

annual ordering, inventory

holding and purchasing

costs

Suppliers’ capacity and

quality constraints

Determine appropriate

level of inventory, number

of orders allocated, size

&frequency of orders

Rezaei &

Davoodi

[20]

MOMINLP

Three objectives of total

profit, inconsistency and

deficiency

Suppliers’ capacity,

budget and storage

capacity constraints

Applying GA (NSGA-II)

to solve model &produce

Pareto-optimal solutions

Senyigit

[22] MILP

Minimizing total cost of

purchasing–production&

distribution strategy

Suppliers’ capacity,

production and storage

capacity constraints

Considering defect rates,

a case study problem

Aliabadi et

al. [23] MINLP

Maximum annual income

(total cost is sum of

suppliers’ and retailer’s

cost)

Suppliers’ capacity and

storage capacity

constraints.

Inventory shortage and

surplus unacceptable

A new two-level GA

Ventura et

al. [24] MINLP

Minimizes total variable

cost including purchasing,

production, inventory and

transportation

Suppliers’ capacity,

production and

distribution capacity, and

inventory capacity

constraints

All-unit discount for

transportation costs

structure

Gorji et al.

[26] MINLP

Maximize total profit

(purchasing, ordering,

inspection, holding,

penalty and transportation)

Suppliers’ capacity and

total capital constraints,

Inventory balance

equations

GA is presented to solve

model

Kang &

Lee [25] MOILP

Minimize total ordering,

holding, purchase and

shortage costs. Maximize

system service level

Inventory balance

equations and shortage

constraints

An efficient heuristic

dynamic programming

(HDP) is

constructed for solving

large-scale stochastic lot-

sizing problems

Meena &

Sarmah [8] MINLP

Minimizing purchasing,

supplier management and

total loss cost

Suppliers’ capacity

constraint

A GA to solve model

considering risk of supply

disruption

lee et al.

[14] MIP

Minimize total ordering,

holding, purchasing,

&transportation cost

Inventory balance

equations

An efficient GA is

proposed to solve model

Choudhary

& Shankar

[11]

MIP

Minimize buyer’s total

cost including purchasing,

transaction, &inventory

cost

Inventory balance

equations ordering cost,

storage and suppliers’

capacity constraint,

Service level requirement

Choudhary

& Shankar

[12]

MOILP

Minimizing net rejected

items, net costs and net

late delivered items

Inventory balance

equations ordering cost,

storage and suppliers’

capacity constraints,

service level requirement

Three variants of goal

programming used to solve

problem

Zaheri et

al. [28]

Mixed binary

NLBLP

First model: Total annual

cost of purchasing, order,

and holding for the leader,

Second model represents

the total annual cost of

production, setup, and

holding for the suppliers.

Production capacity,

Supplier capacity

Using PSO algorithm to

solve their model.

Ghaniabad

i and

Mazinani

[29]

MILP Two models minimizing

the total cost

of ordering costs, the unit

purchase costs, holding

costs and backlogging

costs

Inventory balance

equations,

Using a

commercial optimization

software to solve the

model.

Alfaresa

and

Turnadi

[30]

MIP Minimizes the total cost,

the sum of the

ordering costs, purchasing

cost, transportation cost,

inventory holding cost, and

shortage cost

Inventory balance

constraint,

Transportation capacity

constraints, Budget

limits, Capacity

constraints on storage

volume

Using two heuristic

solution methods: the

Silver-Meal heuristic and

GA

Cardenas-

Barron et

al. [27]

MILP Minimizes

the total purchase cost of

the products, the total

ordering cost, and the total

holding cost for carrying

inventory,

Avoid shortages Heuristic algorithm based

on reduce and optimize

approach (ROA) using

CPLEX to solve the model

Suriyan

and

Rungreung

MILP Minimize total inventory

costs, which consist of

product costs, ordering

Inventory constraints,

Limitation of product

order, Budget limits

Using LINGO to solve

23

anun [31] costs, inventory holding

costs, and transportation

costs

Kanchanar

uangrong

and

Woarawic

hai

[32]

MILP Minimize total costs,

where the costs include

purchase cost of the

products, ordering cost,

transportation cost and

holding cost

Inventory balance

equations, buyer storage

capacity

Using LINGO to solve

This paper MILP

Minimize buyer’s total

cost including purchasing,

ordering, transportation,

inventory and production

costs

Inventory balance

equations,

suppliers’ capacity

constraint, storage

capacity of raw materials

and production constraint

Production costs and

purchase of various raw

materials to produce

products in multi- periods.

GA is used to solve the

model.

Table 2. Data of the raw materials and suppliers’ capacities

i

αip

Wi

CSijt

P=1 P=2 J

1 2 3

1 1 2 2 200 300 500

2 3 1 1 300 250 700

3 2 2 3 250 500 600

Table 3. Manufacturer’s data Product Hpt tp PRpt

1 5 10 10

2 5 12 11

Table 4. Suppliers’ dataset for the numerical example

j Quantity level

Pijkt

I

1 2 3

1

0<Q<100

100≤Q<300

Q≥300

10

8

7

15

14

12

20

18

15

2

0<Q<100

100≤Q<300

Q≥300

11

9

8

17

15

14

19

17

15

3

0<Q<100

100≤Q<300

Q≥300

10

9

8

16

14

13

18

17

16

Table 5. Transportation data

CVjlt UNVlt UVl

J T

1 2 3 1 2 3 4 5

l 1 25 35 45 40 50 50 45 40 20

2 40 50 60 30 60 50 40 35 30

24

Table 6. Results for base example

Objectives Value

Purchasing cost (Z1)

Ordering cost (Z2)

Production cost (Z3)

Inventory holding cost (Z4)

Transportation cost (Z5)

17,050

460

2,650

1,070

3,825

Total Objective 25,055

The amount of raw

materials purchased

Inventory quantities for

raw materials and

products

Quantity of

manufactured

products

The number of

used carriers

X11211=100

X11221=200

X11241=100

X21141=50

X21211=100

X21321=300

X31211=100

X31241=100

X32322=300

I11=20

I21=10

I'12=60

I'13=40

I'14=20

I'22=40

I'23=10

I'24=30

Y11=20

Y12=80

Y21=30

Y22=70

Y24=50

NV111=30

NV112=35

NV114=28

NV222=30

Table 7. The impact of quantity discount

Sub-scenarios

1 2 3

X11111= 85

X21111= 90

X12222= 195

X12242= 120

X32212= 100

X32242= 100

X32322= 300

X23322= 300

X11111=85

X11141=62

X11151=58

X21111=90

X21141=30

X21151=30

X31141=42

X31151=58

X31211=100

X13222= 195

X23322=300

X33322=300

X11111=81

X11121=80

X11131=79

X11141=82

X11151=78

X21111=90

X21121=93

X21131=87

X21141=90

X21151=90

X31131=98

X31151=98

X31211= 100

X31221= 102

X31241= 102

Total cost 26575 27353 27465

Supplier 1

Supplier 2

Supplier 3

~ 18 %

~ 60 %

~ 22 %

~ 41 %

~ 0 %

~ 59 %

100 %

0 %

0 %

25

Table 8. The impact of inventory holding costs

Inventory holding cost for products

(H1t, H2t)

Inventory holding cost for raw materials

(h1t, h2t, h3t)

Scenario 1

(0.5, 0.5) Scenario 2

(10, 10) Scenario 3

(1, 1, 1) Scenario 4

(5, 5, 5) Scenario 5

(13, 13, 13)

X11211= 200

X11233= 200

X21232= 150

X21311= 300

X31232=200

X32312= 300

X11121= 60

X11151=40

X11211=100

X11231=100

X11241=100

X21131=70

X21151= 80

X21211=100

X21221=100,

X21241= 100

X31211= 100,

X31221= 100

X31231= 100,

X31241= 100

X31251= 100

X11211= 200

X11231= 100

X11241= 100

X21131= 50

X21241= 100

X21311= 300

X31231= 100

X31241= 100

X32312= 300

X11211=100

X11221=200

X11241=100

X21141=50

X21211=100

X21321=300

X31211=100

X31241=100

X32322=300

X11111= 80

X11221= 200

X11241= 100

X12122= 20

X21111= 90

X21141= 50

X21321= 300

X22122= 10

X31211= 100

X31241= 100

X32322= 300

I31=20I32=20

I'11=60

I'12=40

I'13=40

I'14=20

I'21=30

I'23=60

I'24=30

I11=20

I13=20

I14=40

I21=10

I22=20

I24=10

I11=20

I12=80

I21=210

I22=120

I23=30

I33=200

I'14=20

I'24=30

I11=20

I21=10

I'12=60

I'13=40

I'14=20

I'22=40

I'23=10

I'24=30

I'12=60

I'13=40

I'14=20

I'22=40

I'23=10

I'24=30

Total Z

Z1

Z2

Z3

Z4

Z5

24,155

17,000

340

2,650

240

3,935

25,830

18,850

600

2,650

280

3,450

24,845

17,000

460

2,650

910

3,825

25,135

17,050

460

2,650

1150

3,825

25,375

17,390

460

2,650

1,000

3,875

26

Table 9. Best value and time for different parameters

Population

size

Maximum

generation

Crossover

rate

Mutation

rate

Elitism

rate

Local

search’s rate

Best Objective

function

Time

(sec.)

250 200 0.93 0.18 0.25 0.2 31167 130

250 200 0.93 0.18 0.1 0.2 31079 135

250 200 0.93 0.1 0.25 0.2 31027 118

250 200 0.85 0.18 0.25 0.2 31007 104

250 200 0.93 0.1 0.1 0.2 30996 81

250 200 0.85 0.18 0.1 0.2 31200 116

250 200 0.85 0.1 0.25 0.2 31072 75

250 200 0.85 0.1 0.1 0.2 31066 85

250 200 0.93 0.18 0.25 0.1 31270 66

250 200 0.93 0.18 0.1 0.1 31210 144

250 200 0.93 0.1 0.25 0.1 31020 112

250 200 0.85 0.18 0.25 0.1 31200 72

250 200 0.93 0.1 0.1 0.1 31039 92

250 200 0.85 0.18 0.1 0.1 31131 95

250 200 0.85 0.1 0.25 0.1 31138 99

250 200 0.85 0.1 0.1 0.1 31162 122

350 150 0.93 0.18 0.25 0.2 30957 119

350 150 0.93 0.18 0.1 0.2 30962 152

350 150 0.93 0.1 0.25 0.2 30930 103

350 150 0.85 0.18 0.25 0.2 31098 113

350 150 0.93 0.1 0.1 0.2 31040 202

350 150 0.85 0.18 0.1 0.2 31070 130

350 150 0.85 0.1 0.25 0.2 31132 104

350 150 0.85 0.1 0.1 0.2 31158 149

350 150 0.93 0.18 0.25 0.1 31017 108

350 150 0.93 0.18 0.1 0.1 31054 118

350 150 0.93 0.1 0.25 0.1 30969 107

350 150 0.85 0.18 0.25 0.1 30988 73

350 150 0.93 0.1 0.1 0.1 31008 134

350 150 0.85 0.18 0.1 0.1 31146 93

350 150 0.85 0.1 0.25 0.1 31186 99

350 150 0.85 0.1 0.1 0.1 31037 105

Table 10. Samples’ dimensions Sample

Raw

material

(i)

Supplier

(j)

Break

levels

(k)

Product

(p)

Period

(t)

Carrier

(l)

Considerations

Small-

scale

1 1 2 3 1 2 2 j1, j2

2 1 2 3 1 3 2 j1, j2

3 2 2 3 2 3 2 j1, j2

4 3 2 3 1 3 2 j2, j3

5 3 3 3 1 3 2 ---

6 3 3 3 2 3 2 ---

7 2 2 3 2 5 2 j1, j2

8 3 3 3 2 4 2 ---

9 3 3 3 2 5 2 ---

10 3 3 3 2 6 2 ---

Large-

scale

11 3 3 3 2 10 2 ---

12 3 5 3 2 9 2 ---

13 4 3 3 3 10 2 ---

14 3 5 3 3 10 3 ---

15 4 10 3 3 13 3 ---

16 3 12 3 3 15 3 ---

17 3 10 3 3 15 3 ---

18 3 15 3 3 10 3 ---

19 3 15 3 3 15 3 ---

20 3 15 3 3 20 3 ---

27

Table 11. The values of the parameters used in the samples Notes Values Dpt ~ uni [20, 50] Ojt ~ uni [80, 150] Pijkt ~ uni [10, 30] hit ~ uni [2, 5] Hpt ~ uni [4, 9] PRpt ~ uni [10, 20] CVjlt ~ uni [50 , 90] Bijkt 100, 300, 1000 CRt ~ uni [800, 1500] CPt ~ uni [100, 200] CSijt ~ uni [300, 900] tp ~ uni [2, 5] Tt ~ uni [400, 800] αip ~ uni [1, 5] Wi ~ uni [1, 7] UVl ~ uni [50, 100] UNVlt ~ uni [30, 60]

Table 12. The computational results of small-scale samples

Sample

LINGO Genetic Algorithm (GA)

Gap

(%) Run

time(sec.)

Optimal

solution

Run time

(average of 5 runs)

Solution value

(average of 5 runs)

Best solution

value

1 1 1,600 11 1,600 1,600 0.00

2 1 2,220 13 2,230 2,230 0.45

3 2 8,780 34 8,828.4 8,806 0.29

4 3 7,470 34 7,623 7,616 1.95

5 5 7,125 27 7,239 7,235 1.54

6 5 15,240 47 15,393 15,347 0.70

7 7 14,515 53 14,862.4 14,763 1.70

8 20 19,925 62 20,663.3 20,613 3.45

9 134 24,731 87 25,684 25,561 3.35

10 85 30,230 119 30,997 30,930 2.3

Table 13. The computational results of the large-scale samples

Sample LINGO Genetic Algorithm (GA)

Gap

% Run time

(sec.)

Best

solution

Run time

(average of 5 runs)

Solution value

(average of 5 runs)

Best solution

value

11 2,000 119,123 284 122,886 122440 2.78

12 2,000 78,885.4 224 82,118.4 81384 3.16

13 2,000 168,052 283 170,238 169740 1.00

14 2,000 138,988 231 143,006 141980 2.15

15 2,000 167,819 541 174,824 173940 3.64

16 2,000 171,049 448 179,408 177200 3.59

17 2,000 207,025 427 220,880 218030 5.31

18 2,000 104,101 277 112,112 109710 5.38

19 4,000 208,993 437 225,602 223220 6.80

20 4,000 250,818 390 262,856 260390 3.81